Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is

Given information:

• Distance AB = 45 km

• They meet after 1 hour 30 minutes = 1.5 hours

• Anil reaches B exactly 1 hour 15 minutes = 1.25 hours after Sunil reaches A

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When two people start from opposite ends and meet somewhere in between, there's a beautiful relationship we can use. Let us explain this with an example first:

Imagine Anil and Sunil meet at point C. After meeting:

• Anil still needs time $t_A$ to reach B

• Sunil still needs time $t_S$ to reach A

Here's the key insight: The distance Anil covers in 1.5 hours = The distance Sunil covers in time $t_S$

Similarly: The distance Sunil covers in 1.5 hours = The distance Anil covers in time $t_A$

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Let's look at this mathematically:

Since $s=\frac{d}{t}$

Here $t=1.5$ and distance is the same, we can write algebraly:

$\begin{aligned} & s_a \times 1.5=s_s t_s \\ & s_a t_a=s_s \times 1.5 \\ \\ &\text{put }s_a=\frac{s_s t_s}{1.5} \quad \text { in equation (2) } \\ \Rightarrow & \frac{s_s t_s}{1.5} \cdot t_{A}=s_s \times 1.5 \end{aligned}$

Hence, $t_s.t_{a}=(1.5)^{2}$

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From this relationship, we can derive that:

$\text{Meeting time} = \sqrt{t_A \times t_S}$

This would hold true for all questions of this kind.

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We know Anil reaches B exactly 1.25 hours after Sunil reaches A:

$t_A = t_S + 1.25$

Substituting into our meeting formula:

$1.5 = \sqrt{(t_S + 1.25) \times t_S}$

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$(1.5)^2 = (t_S + 1.25) \times t_S$

$2.25 = t_S^2 + 1.25t_S$

Remove decimals:

$9 = 4t_S^2 + 5t_S$

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To factor $4t_S^2 + 5t_S - 9 = 0$, we'll look for two numbers that multiply to give $(4)(-9) = -36$ and add to give $5$.

$4t_S^2 + 9t_S - 4t_S - 9 = 0$

$t_S(4t_S + 9) - 1(4t_S + 9) = 0$

$(t_S - 1)(4t_S + 9) = 0$

This gives us $t_S = 1$ or $t_S = -\tfrac{9}{4}$

Since time cannot be negative: $t_S = 1$ hour

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$t_A = t_S + 1.25 = 1 + 1.25 = 2.25$ hours

Anil's total journey time = Meeting time + Time after meeting

$= 1.5 + 2.25 = 3.75$ hours

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$\text{Speed} = \tfrac{\text{Distance}}{\text{Time}} = \tfrac{45}{3.75} = 12 \text{ km/h}$

Therefore, Anil's speed is 12 km/h.